# Cunningham function

In statistics, the Cunningham function or Pearson–Cunningham function ωm,n(x) is a generalisation of a special function introduced by (Pearson 1906) and studied in the form here by (Cunningham 1908). It can be defined in terms of the confluent hypergeometric function U, by

$\displaystyle{ \displaystyle \omega_{m,n}(x) = \frac{e^{-x+\pi i (m/2-n)}}{\Gamma(1+n-m/2)}U(m/2-n,1+m,x). }$

The function was studied by Cunningham[1] in the context of a multivariate generalisation of the Edgeworth expansion for approximating a probability density function based on its (joint) moments. In a more general context, the function is related to the solution of the constant-coefficient diffusion equation, in one or more dimensions.[1]

The function ωm,n(x) is a solution of the differential equation for X:[1]

$\displaystyle{ xX''+(x+1+m)X'+(n+\tfrac{1}{2}m+1)X. }$

The special function studied by Pearson is given, in his notation by,[1]

$\displaystyle{ \omega_{2n}(x) =\omega_{0,n}(x). }$

## Notes

1. (Cunningham 1908)

## References

• Abramowitz, Milton; Stegun, Irene Ann, eds (1983). "Chapter 13". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 510. LCCN 65-12253. ISBN 978-0-486-61272-0.
• Cunningham, E. (1908), "The ω-Functions, a Class of Normal Functions Occurring in Statistics", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (The Royal Society) 81 (548): 310–331, doi:10.1098/rspa.1908.0085, ISSN 0950-1207
• Pearson, Karl (1906), A mathematical theory of random migration, London, Dulau and co.
• Whittaker, E. T.; Watson, G. N. (1963), A Course in Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2  See exercise 10, chapter XVI, p. 353